This paper is devoted to the study of a class of optimal control problems governed by 1–D Kobayashi–Warren–Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141–150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi–Warren–Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1–3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.

中文翻译：

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一维Kobayashi–Warren–Carter型系统控制的最优控制问题

本文致力于研究由一维Kobayashi-Warren-Carter型系统控制的一类最优控制问题，该系统基于[Kobayashi等人，Physica D ，140，141–150，2000]。该类包括Kobayashi–Warren–Carter型物理现实状态系统的最优控制问题及其正则逼近问题。本文的结果在三个主要定理1-3中陈述。第一个主定理1与状态系统的可解性和连续依赖性有关。同时，第二个主定理2涉及最优控制问题的可解性，以及我们的最优控制问题中的一些半连续关联。最后，在第三个主定理3中，我们推导了正则化近似问题的最优控制的一阶必要最优性条件。通过采用近似极限，我们还可以得出针对物理现实问题的最优控制的最优条件。